3.2255 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=118 \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-2 b e g-5 c d g+9 c e f)}{63 c^2 e^2 (d+e x)^{7/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2 (d+e x)^{5/2}} \]

[Out]

(-2*(9*c*e*f - 5*c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(
63*c^2*e^2*(d + e*x)^(7/2)) - (2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/
(9*c*e^2*(d + e*x)^(5/2))

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Rubi [A]  time = 0.480894, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-2 b e g-5 c d g+9 c e f)}{63 c^2 e^2 (d+e x)^{7/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(5/2),x]

[Out]

(-2*(9*c*e*f - 5*c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(
63*c^2*e^2*(d + e*x)^(7/2)) - (2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/
(9*c*e^2*(d + e*x)^(5/2))

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Rubi in Sympy [A]  time = 44.7816, size = 110, normalized size = 0.93 \[ - \frac{2 g \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{9 c e^{2} \left (d + e x\right )^{\frac{5}{2}}} + \frac{2 \left (2 b e g + 5 c d g - 9 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{63 c^{2} e^{2} \left (d + e x\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(5/2),x)

[Out]

-2*g*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(9*c*e**2*(d + e*x)**(5/2
)) + 2*(2*b*e*g + 5*c*d*g - 9*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))*
*(7/2)/(63*c**2*e**2*(d + e*x)**(7/2))

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Mathematica [A]  time = 0.190484, size = 78, normalized size = 0.66 \[ \frac{2 (b e-c d+c e x)^3 \sqrt{(d+e x) (c (d-e x)-b e)} (c (2 d g+9 e f+7 e g x)-2 b e g)}{63 c^2 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(5/2),x]

[Out]

(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-2*b*e*g + c
*(9*e*f + 2*d*g + 7*e*g*x)))/(63*c^2*e^2*Sqrt[d + e*x])

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Maple [A]  time = 0.007, size = 79, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -7\,cegx+2\,beg-2\,cdg-9\,fce \right ) }{63\,{c}^{2}{e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(5/2),x)

[Out]

-2/63*(c*e*x+b*e-c*d)*(-7*c*e*g*x+2*b*e*g-2*c*d*g-9*c*e*f)*(-c*e^2*x^2-b*e^2*x-b
*d*e+c*d^2)^(5/2)/c^2/e^2/(e*x+d)^(5/2)

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Maxima [A]  time = 0.729761, size = 424, normalized size = 3.59 \[ \frac{2 \,{\left (c^{3} e^{3} x^{3} - c^{3} d^{3} + 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + b^{3} e^{3} - 3 \,{\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} + 3 \,{\left (c^{3} d^{2} e - 2 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x\right )} \sqrt{-c e x + c d - b e} f}{7 \, c e} + \frac{2 \,{\left (7 \, c^{4} e^{4} x^{4} - 2 \, c^{4} d^{4} + 8 \, b c^{3} d^{3} e - 12 \, b^{2} c^{2} d^{2} e^{2} + 8 \, b^{3} c d e^{3} - 2 \, b^{4} e^{4} - 19 \,{\left (c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{3} + 15 \,{\left (c^{4} d^{2} e^{2} - 2 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4}\right )} x^{2} -{\left (c^{4} d^{3} e - 3 \, b c^{3} d^{2} e^{2} + 3 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} x\right )} \sqrt{-c e x + c d - b e} g}{63 \, c^{2} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

2/7*(c^3*e^3*x^3 - c^3*d^3 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + b^3*e^3 - 3*(c^3*d*
e^2 - b*c^2*e^3)*x^2 + 3*(c^3*d^2*e - 2*b*c^2*d*e^2 + b^2*c*e^3)*x)*sqrt(-c*e*x
+ c*d - b*e)*f/(c*e) + 2/63*(7*c^4*e^4*x^4 - 2*c^4*d^4 + 8*b*c^3*d^3*e - 12*b^2*
c^2*d^2*e^2 + 8*b^3*c*d*e^3 - 2*b^4*e^4 - 19*(c^4*d*e^3 - b*c^3*e^4)*x^3 + 15*(c
^4*d^2*e^2 - 2*b*c^3*d*e^3 + b^2*c^2*e^4)*x^2 - (c^4*d^3*e - 3*b*c^3*d^2*e^2 + 3
*b^2*c^2*d*e^3 - b^3*c*e^4)*x)*sqrt(-c*e*x + c*d - b*e)*g/(c^2*e^2)

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Fricas [A]  time = 0.28724, size = 817, normalized size = 6.92 \[ -\frac{2 \,{\left (7 \, c^{5} e^{6} g x^{6} +{\left (9 \, c^{5} e^{6} f -{\left (19 \, c^{5} d e^{5} - 26 \, b c^{4} e^{6}\right )} g\right )} x^{5} -{\left (9 \,{\left (3 \, c^{5} d e^{5} - 4 \, b c^{4} e^{6}\right )} f - 2 \,{\left (4 \, c^{5} d^{2} e^{4} - 21 \, b c^{4} d e^{5} + 17 \, b^{2} c^{3} e^{6}\right )} g\right )} x^{4} + 2 \,{\left (9 \,{\left (c^{5} d^{2} e^{4} - 4 \, b c^{4} d e^{5} + 3 \, b^{2} c^{3} e^{6}\right )} f +{\left (9 \, c^{5} d^{3} e^{3} - 10 \, b c^{4} d^{2} e^{4} - 7 \, b^{2} c^{3} d e^{5} + 8 \, b^{3} c^{2} e^{6}\right )} g\right )} x^{3} +{\left (18 \,{\left (c^{5} d^{3} e^{3} - 3 \, b^{2} c^{3} d e^{5} + 2 \, b^{3} c^{2} e^{6}\right )} f -{\left (17 \, c^{5} d^{4} e^{2} - 52 \, b c^{4} d^{3} e^{3} + 54 \, b^{2} c^{3} d^{2} e^{4} - 20 \, b^{3} c^{2} d e^{5} + b^{4} c e^{6}\right )} g\right )} x^{2} + 9 \,{\left (c^{5} d^{5} e - 4 \, b c^{4} d^{4} e^{2} + 6 \, b^{2} c^{3} d^{3} e^{3} - 4 \, b^{3} c^{2} d^{2} e^{4} + b^{4} c d e^{5}\right )} f + 2 \,{\left (c^{5} d^{6} - 5 \, b c^{4} d^{5} e + 10 \, b^{2} c^{3} d^{4} e^{2} - 10 \, b^{3} c^{2} d^{3} e^{3} + 5 \, b^{4} c d^{2} e^{4} - b^{5} d e^{5}\right )} g -{\left (9 \,{\left (3 \, c^{5} d^{4} e^{2} - 8 \, b c^{4} d^{3} e^{3} + 6 \, b^{2} c^{3} d^{2} e^{4} - b^{4} c e^{6}\right )} f -{\left (c^{5} d^{5} e - 6 \, b c^{4} d^{4} e^{2} + 14 \, b^{2} c^{3} d^{3} e^{3} - 16 \, b^{3} c^{2} d^{2} e^{4} + 9 \, b^{4} c d e^{5} - 2 \, b^{5} e^{6}\right )} g\right )} x\right )}}{63 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d} c^{2} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

-2/63*(7*c^5*e^6*g*x^6 + (9*c^5*e^6*f - (19*c^5*d*e^5 - 26*b*c^4*e^6)*g)*x^5 - (
9*(3*c^5*d*e^5 - 4*b*c^4*e^6)*f - 2*(4*c^5*d^2*e^4 - 21*b*c^4*d*e^5 + 17*b^2*c^3
*e^6)*g)*x^4 + 2*(9*(c^5*d^2*e^4 - 4*b*c^4*d*e^5 + 3*b^2*c^3*e^6)*f + (9*c^5*d^3
*e^3 - 10*b*c^4*d^2*e^4 - 7*b^2*c^3*d*e^5 + 8*b^3*c^2*e^6)*g)*x^3 + (18*(c^5*d^3
*e^3 - 3*b^2*c^3*d*e^5 + 2*b^3*c^2*e^6)*f - (17*c^5*d^4*e^2 - 52*b*c^4*d^3*e^3 +
54*b^2*c^3*d^2*e^4 - 20*b^3*c^2*d*e^5 + b^4*c*e^6)*g)*x^2 + 9*(c^5*d^5*e - 4*b*
c^4*d^4*e^2 + 6*b^2*c^3*d^3*e^3 - 4*b^3*c^2*d^2*e^4 + b^4*c*d*e^5)*f + 2*(c^5*d^
6 - 5*b*c^4*d^5*e + 10*b^2*c^3*d^4*e^2 - 10*b^3*c^2*d^3*e^3 + 5*b^4*c*d^2*e^4 -
b^5*d*e^5)*g - (9*(3*c^5*d^4*e^2 - 8*b*c^4*d^3*e^3 + 6*b^2*c^3*d^2*e^4 - b^4*c*e
^6)*f - (c^5*d^5*e - 6*b*c^4*d^4*e^2 + 14*b^2*c^3*d^3*e^3 - 16*b^3*c^2*d^2*e^4 +
9*b^4*c*d*e^5 - 2*b^5*e^6)*g)*x)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sq
rt(e*x + d)*c^2*e^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(5/2),x, algorithm="giac")

[Out]

Timed out